Category: Knowledge

One of the most frustrating things that I regularly encounter is clearly intelligent people demeaning themselves. One of the most tragic is seeing clearly intelligent children told they are not good at something. In The Geography of Thought Richard Nesbit highlights the particularly western perception that people have certain fixed attributes and competencies whether they be sociability, musicality, writing ability, mathematics or any number of other things. This is for the most part just wrong. Many fail to appreciate the degree to which people able to change themselves; our personalities and aptitudes are constantly developing. Once we acknowledge that we are inherently changeable we are able to see that what is more important than talent is just plain work. No one wakes up one day able to play piano, solve differential calculus equations or paint, or write poetry. People are able to do these things because they devote a huge amount of time and effort to them. It’s horrendously unfair to attribute these achievements merely to talent.

As someone with eclectic interests, a principle that I live my life by is that with some effort I can generally achieve a reasonable degree of proficiency in most things. This is how, after struggling through mathematics in high school and generally considering myself of at best average ability, I was able to complete two university courses in pure mathematics with relative success.

So what turned things around? I think I owe a large part of my success to my father’s ceaseless, to the point of being infuriating, insistence that I was good at maths despite my bad results in school. At the time I thought of a number of reasons why he would say this. Principally I thought he was just assuming that because he was good at maths that I would be too. Whatever the reason, he said it enough that even though I achieved mediocre results in school I eventually believed him.

In addition to giving me the confidence to attempt mathematics at a later stage this forced me to locate the reason for my poor results somewhere other than an innate lack of talent. And there was only one thing to blame, really. I was lazy. From then on I had the belief that I was probably okay at maths as long as I applied myself. Still I held this belief for almost two years before I did anything about it.

A semester before I started studying mathematics again I took a course in introductory logic under the philosophy department. It was a difficult course but I assumed that I would do well because I “knew” I was good at philosophy. As it turned out, I did excel, but at the same time I realised that what I was doing was essentially mathematics. And I loved it. The surprise of finding hidden contradictions and unexpected but irrefutable conclusions from seemingly innocuous premises was intoxicating. I realised this by approaching the subject with confidence and positive belief. This is what finally prompted me to take up mathematics for the first time since high school.

Now I don’t want to give the impression that it was smooth sailing from there on out, because it certainly wasn’t. I wasn’t able to pick a course out of nowhere, in a field that was for the most part foreign to me, and succeed just because I was “clever”. Truth be told the first course I took in Discrete mathematics (not discreet) was worth 3 out of 24 credit points for the semester and it probably took up about half of my time. I remember spending a disproportionately large number of hours on an assignment that was only worth 5% of my total mark for the course only to get 2/5 for it. That was definitely discouraging but I found consolation in the fact that I had lost one mark out of sheer carelessness (I had erroneously written in my last line of working that 2^8 equals 64) and I persevered to pass the course comfortably.

Even now I don’t see myself as particularly exceptional in mathematics but I’m also aware that the people I am comparing myself to have spent an inordinately greater amount of time and effort on the subject than I have. Two semesters at university is really not that much. Nevertheless this was a life changing experience, I had turned what I considered a great weakness into a strength, and now I’m able to help others achieve the same thing.

The benefits of this experience reach far beyond proficiency in mathematics. Since then I’ve never limited myself to anything. Just this year I’ve taken up piano and learning German, both of which I am now competent in to a degree that I honestly never expected. Learning a foreign language especially was something that I had given up on in high school after lacklustre experiences in German and Japanese (which were mostly my fault).

Having said all this, I do not wish to imply that a struggling student struggles simply because she does not try hard enough. Effort directed towards the wrong area is practically useless. Often it is harder to determine where and how effort should be applied than to spend hours aimlessly slaving over a text-book. The point is that for the majority of students a lack of talent is not what is holding them back, and the more that they believe that it is, the more they allow themselves to be lazy. But this is a good thing! It’s easy to address. Hard work and a positive attitude may seem like tired advice to give but it is inevitably overlooked.

I also don’t want to lead anyone to believe that it is necessary to set ones goals at the same level as I do. I set my goals at the level I do because I know what motivations me. I know that I’m not discouraged by striving for things which I’m unsure of whether I can achieve. But I also know that there are many people for whom this is not the case. Indeed some people are more productive with sequential easily achievable goals. To set a highly challenging goal at the outset would paralyse them. Despite this, what I do want to get across is that most students are capable of far more than they believe or are told, and for this reason they should set their goals accordingly, even if it is just a little higher. If we work towards being fearless of failure, soon it becomes clear that the higher we set our goals, the more we allow ourselves to achieve, regardless of whether we reach our goal or not.

It would be naive to assert that talent doesn’t exist. But by definition, it’s innate and unchangeable, so it’s useless to concern ourselves with it. We are much better of focussing on what we can achieve, which for most of us is a very great deal. Even for those who might be called talented, they didn’t get to where they are by talent alone. Setting challenging goals and positively pursuing them is the best way to become more competent, regardless of talent.

Something I’ve often thought about is how the analysis of music relates to the way in which we experience music and its constituent elements. Particularly I’m interested in the extent that certain, arguably, inaudible structures can be said to exist. Even more so I find fascinating those elements of music that while not initially apparent reveal themselves with either familiarity or close study.

Few uninitiated listeners could hope to hear, for example, each distinct voice in a complex polyphonic passage of music, but with training one learns to differentiate voices with relative ease. Now we would all agree that the music does in fact consist of multiple, essentially solo melodies sounded simultaneously. This phenomenon becomes more nuanced as we consider more opaque musical structures such as the permutations of a tone row, which range from relatively easy to perceive transformations such as augmentation, diminution and transposition to more extreme transformations, such as suppletion and the arranging of elements vertically rather than horizontally, which often result in a permutation that seems entirely distinct from the original.

There is often debate over whether these processes are in fact audible, principally, it seems, because audibility is considered a prerequisite to being worthy of study. I think that the relationship between how we hear music and analysis can be made clearer using, for arguments sake, the idea of a musical ‘structure’.

What is a musical structure?

For the purposes of this post I will define a musical structure as some sort of relationship between one discrete musical element or group of elements with another in a certain musical dimension. So a structure can be a harmony, the simultaneous sounding of two or more notes; a melody, two or more notes played in sequence; a rhythm, two or more notes played at a certain time interval; or the organisation of different timbres/tone colours. These are but a few of the more fundamental structures that can be said to exist in music, all of them are easily audible and rarely ambiguous.

Ambiguity is created when elements can be interpreted as part of more than one structure. As structures become more complex, usually involving a combination of fundamental structures in some form, they become less audible and the potential for ambiguity is increased.

What it is to be ‘audible’?

To be audible is essentially just to be among the simpler, direct structures. Complex structures will always have more simple intervening structures, where there are enough of these, of certain complexity, our ability to perceive the underlying structure is diminished, eventually to the point that we are oblivious too them.

However sometimes we are able to turn our attention away from the surface structures and perceive the deeper structures in the same way that one may decipher a code by selectively reading words from a text.

Are the structures inherently audible in music or projected by analysis upon it?

Subtle aural structures may not always impress themselves upon the listener but may rather be projected by the listener onto the music. Nonetheless I think that generally analysis serves to bring what already exists in the music into focus.

While music is primarily experienced through sound, there are other mediums through which to appreciate it much as one appreciates a painting through is aesthetic beauty, narrative, symbolism, emotional impact and historical context among a myriad of other factors while it remains a primarily visual meaning. It is conceivable that in music and in visual art that these modes of appreciation are not wholly communicated through the artwork itself but also by secondary, associated artefacts, such as scores, programs, critiques and the lexicon of previously defined gestures which are easily accidentally employed i.e. a subtle cross in a painting may not always be an allusion to Christ and more pertinently, music similarities do not necessarily correlate with semantic ones.

This type of speculation could go on indefinitely. Perhaps it is more convenient to consider the sound element of a work is the only part that can be properly called music and everything else becomes part of a subordinate, though inextricable, non-music artwork. Even if this is done, we are brought back to the question of what constitutes audibility.

Does this add and explanatory power? Or allow a more precise definition of music?

It is relatively easy to see the significance of structures, inaudible though they may be, that add semantic content to a piece of music. What is less obvious is whether inaudible structures such as the forms of self imitation already described, which add no obvious semantic or otherwise content, are valuable. Personally I feel there is beauty in these structures, even if I don’t experience it aurally. If this is the case then clearly it is no longer necessary for them to add content, but in what sense are they music. These structures may not be perceived aurally but nonetheless they do manifest as sound.